The equivalent local form of (REF ) can be obtained using [1]}, in addition to Identities REF  and REF , as
\(D\Delta ^2 w - [\phi ,w]=f_1+ D\left( (1-\nu )\operatorname{div}\operatorname{div}{\lambda ^p} +\nu \Delta (\operatorname{tr}({\lambda ^p}))\right) ~ \text{in}~\Omega -\lbrace S\cup \lbrace O\rbrace \rbrace ,\\\begin{aligned}D(1-\nu )\frac{d^2}{ds^2} \left\langle \llbracket \nabla w\rrbracket , \nu  \right\rangle + D\left\langle \llbracket \operatorname{div}\nabla \nabla w \rrbracket , \nu  \right\rangle - \left\langle \nabla \nabla \phi , t\otimes t \right\rangle \left\langle \llbracket \nabla w\rrbracket , \nu  \right\rangle = f_2 + \\D(1-\nu )\left(\left\langle \llbracket \operatorname{div}\lambda ^p \rrbracket , \nu  \right\rangle + \left\langle \frac{d}{ds} \llbracket \lambda ^p\rrbracket ,t\otimes \nu  \right\rangle + k\langle \llbracket \lambda ^p\rrbracket ,s \rangle \right) + D\nu \llbracket \nabla (\operatorname{tr}\lambda ^p) \rrbracket ~\text{on} ~S-\lbrace O\rbrace ,\end{aligned}\\(1-\nu )k \left\langle \llbracket \nabla w\rrbracket , \nu  \right\rangle + \llbracket \Delta w\rrbracket = (1-\nu ) \langle \llbracket \lambda ^p\rrbracket ,\nu \otimes \nu  \rangle +\nu \llbracket \operatorname{tr}(\lambda ^p)\rrbracket ~\text{on} ~S-\lbrace O\rbrace ,~\text{and}\\D W(\Delta ^2 v^\alpha )+\nabla \Phi \otimes \nabla W(\mathbb {A}\nabla \nabla v^\alpha )=F(v^\alpha ) -D\left( (1-\nu )\Lambda ^p(\nabla \nabla v^\alpha ) +\nu (\operatorname{tr}(\Lambda ^p)(\Delta v^\alpha ))\right)~\text{at}~O,\) 
